Generalized Ordered Logit Models

1. Generalized Ordered Logit Models Richard Williams University of Notre Dame, Department of Sociology rwilliam@ND.Edu Annual Midwest Sociological Society Meetings April 2, 2010

2. Introduction We are used to estimating models where a continuous dependent variable, Y, is regressed on an independent variable, X But suppose the observed Y is not continuous � instead, it is a collapsed version of an underlying unobserved variable, Y*

3. Examples: Income, coded in categories like $0 = 1, $1- $10,000 = 2, $10,001-$30,000 = 3, $30,001-$60,000 = 4, $60,001 or higher = 5 Do you approve or disapprove of the President's health care plan? 1 = Strongly disapprove, 2 = Disapprove, 3 = Approve, 4 = Strongly approve.

4. For such variables, also known as limited dependent variables, we know the interval that the underlying Y* falls in, but not its exact value. Ordinal regression techniques allow us to estimate the effects of the Xs on the underlying Y*.

5. Example: Ordered logit model (Adapted from Long & Freese, 2003 � Data from the 1977 & 1989 General Social Survey) Respondents are asked to evaluate the following statement: �A working mother can establish just as warm and secure a relationship with her child as a mother who does not work.� 1 = Strongly Disagree (SD) 2 = Disagree (D) 3 = Agree (A) 4 = Strongly Agree (SA).

6. Explanatory variables are yr89 (survey year; 0 = 1977, 1 = 1989) male (0 = female, 1 = male) white (0 = nonwhite, 1 = white) age (measured in years) ed (years of education) prst (occupational prestige scale).

7. Ologit results . ologit warm yr89 male white age ed prst Ordered logit estimates Number of obs = 2293 LR chi2(6) = 301.72 Prob > chi2 = 0.0000 Log likelihood = -2844.9123 Pseudo R2 = 0.0504 ------------------------------------------------------------------------------ warm | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- yr89 | .5239025 .0798988 6.56 0.000 .3673037 .6805013 male | -.7332997 .0784827 -9.34 0.000 -.8871229 -.5794766 white | -.3911595 .1183808 -3.30 0.001 -.6231815 -.1591374 age | -.0216655 .0024683 -8.78 0.000 -.0265032 -.0168278 ed | .0671728 .015975 4.20 0.000 .0358624 .0984831 prst | .0060727 .0032929 1.84 0.065 -.0003813 .0125267 -------------+---------------------------------------------------------------- _cut1 | -2.465362 .2389126 (Ancillary parameters) _cut2 | -.630904 .2333155 _cut3 | 1.261854 .2340179 ------------------------------------------------------------------------------

8. Brant test shows assumptions violated . brant Brant Test of Parallel Regression Assumption Variable | chi2 p>chi2 df -------------+-------------------------- All | 49.18 0.000 12 -------------+-------------------------- yr89 | 13.01 0.001 2 male | 22.24 0.000 2 white | 1.27 0.531 2 age | 7.38 0.025 2 ed | 4.31 0.116 2 prst | 4.33 0.115 2 ---------------------------------------- A significant test statistic provides evidence that the parallel regression assumption has been violated.

9. How are the assumptions violated? . brant, detail Estimated coefficients from j-1 binary regressions y>1 y>2 y>3 yr89 .9647422 .56540626 .31907316 male -.30536425 -.69054232 -1.0837888 white -.55265759 -.31427081 -.39299842 age -.0164704 -.02533448 -.01859051 ed .10479624 .05285265 .05755466 prst -.00141118 .00953216 .00553043 _cons 1.8584045 .73032873 -1.0245168 This is a series of binary logistic regressions. First it is 1 versus 2,3,4; then 1 & 2 versus 3 & 4; then 1, 2, 3 versus 4 If proportional odds/ parallel lines assumptions were not violated, all of these coefficients (except the intercepts) would be the same except for sampling variability.

10. Example of when assumptions are not violated

11. Examples of how assumptions can be violated

12. Examples of how assumptions can be violated

13. Examples of how assumptions can be violated

14. Every one of the above models represents a reasonable relationship involving an ordinal variable; but only the proportional odds model does not violate the assumptions of the ordered logit model FURTHER, there could be a dozen variables in a model, 11 of which meet the proportional odds assumption and only one of which does not We therefore want a more flexible and parsimonious model that can deal with situations like the above

15. Unconstrained gologit model Unconstrained gologit results are very similar to what we get with the series of binary logistic regressions and can be interpreted the same way. The gologit model can be written as

16. The ologit model is a special case of the gologit model, where the betas are the same for each j (NOTE: ologit actually reports cut points, which equal the negatives of the alphas used here)

17. Partial Proportional Odds Model A key enhancement of gologit2 is that it allows some of the beta coefficients to be the same for all values of j, while others can differ. i.e. it can estimate partial proportional odds models. For example, in the following the betas for X1 and X2 are constrained but the betas for X3 are not.

18. Either mlogit or unconstrained gologit can be overkill � both generate many more parameters than ologit does. All variables are freed from the proportional odds constraint, even though the assumption may only be violated by one or a few of them gologit2, with the autofit option, will only relax the parallel lines constraint for those variables where it is violated

20. Interpretation Once we have the results though, how do we interpret them??? There are several possibilities.

21. Interpretation 1: The effect of x on y depends on the value of y Our earlier proportional odds examples show how this could plausibly be true Hedeker and Mermelstein (1998) also raise the idea that the categories of the DV may represent stages, e.g. pre-contemplation, contemplation, and action. An intervention might be effective in moving people from pre-contemplation to contemplation, but be ineffective in moving people from contemplation to action. If so, the effects of an explanatory variable will not be the same across the K-1 cumulative logits of the model

22. Working mother�s example Effects of the constrained variables (white, age, ed, prst) can be interpreted pretty much the same as they were in the earlier ologit model. For yr89 and male, the differences from before are largely just a matter of degree. People became more supportive of working mothers across time, but the greatest effect of time was to push people away from the most extremely negative attitudes. For gender, men were less supportive of working mothers than were women, but they were especially unlikely to have strongly favorable attitudes.

23. Substantive example: Boes & Winkelman, 2004:�Completely missing so far is any evidence whether the magnitude of the income effect depends on a person�s happiness: is it possible that the effect of income on happiness is different in different parts of the outcome distribution? Could it be that �money cannot buy happiness, but buy-off unhappiness� as a proverb says? And if so, how can such distributional effects be quantified?�

24. Interpretation 2: State-dependent reporting bias - gologit as measurement model As noted, the idea behind y* is that there is an unobserved continuous variable that gets collapsed into the limited number of categories for the observed variable y. HOWEVER, respondents have to decide how that collapsing should be done, e.g. they have to decide whether their feelings cross the threshold between �agree� and �strongly agree,� whether their health is �good� or �very good,� etc.

25. Respondents do NOT necessarily use the same frame of reference when answering, e.g. the elderly may use a different frame of reference than the young do when assessing their health Other factors can also cause respondents to employ different thresholds when describing things Some groups may be more modest in describing their wealth, IQ or other characteristics

26. In these cases the underlying latent variable may be the same for all groups; but the thresholds/cut points used may vary. Example: an estimated gender effect could reflect differences in measurement across genders rather than a real gender effect on the outcome of interest. Lindeboom & Doorslaer (2004) note that this has been referred to as state-dependent reporting bias, scale of reference bias, response category cut-point shift, reporting heterogeneity & differential item functioning.

27. If the difference in thresholds is constant (index shift), proportional odds will still hold EX: Women�s cutpoints are all a half point higher than the corresponding male cutpoints ologit could be used in such cases If the difference is not constant (cut point shift), proportional odds will be violated EX: Men and women might have the same thresholds at lower levels of pain but have different thresholds for higher levels A gologit/ partial proportional odds model can capture this

28. If you are confident that some apparent effects reflect differences in measurement rather than real differences in effects, then Cutpoints (and their determinants) are substantively interesting, rather than just �nuisance� parameters The idea of an underlying y* is preserved (Determinants of y* are the same for all, but cutpoints differ across individuals and groups)

29. Key advantage: This could greatly improve cross-group comparisons, getting rid of artifactual differences caused by differences in measurement. Key Concern: Can you really be sure the coefficients reflect measurement and not real effects, or some combination of real & measurement effects?

30. Theory may help � if your model strongly claims the effect of gender should be zero, then any observed effect of gender can be attributed to measurement differences. But regardless of what your theory says, you may at least want to acknowledge the possibility that apparent effects could be �real� or just measurement artifacts.

31. Interpretation 3: The outcome ismulti-dimensional A variable that is ordinal in some respects may not be ordinal or else be differently-ordinal in others. E.g. variables could be ordered either by direction (Strongly disagree to Strongly Agree) or intensity (Indifferent to Feel Strongly)

32. Suppose women tend to take less extreme political positions than men. Using the first (directional) coding, an ordinal model might not work very well, whereas it could work well with the 2nd (intensity) coding. But, suppose that for every other independent variable the directional coding works fine in an ordinal model.

33. Our choices in the past have either been to (a) run ordered logit, with the model really not appropriate for the gender variable, or (b) run multinomial logit, ignoring the parsimony of the ordinal model just because one variable doesn�t work with it. With gologit models, we have option (c) � constrain the vars where it works to meet the parallel lines assumption, while freeing up other vars (e.g. gender) from that constraint.

34. Interpretation 4: gologit as non-linear probability model As Long & Freese (2006, p. 187) point out �The ordinal regression model can also be developed as a nonlinear probability model without appealing to the idea of a latent variable.� Ergo, the simplest thing may just be to interpret gologit as a non-linear probability model that lets you estimate the determinants & probability of each outcome occurring. Forget about the idea of a y* Other interpretations, such as we have just discussed, can preserve or modify the idea of an underlying y*

35. For more information, see: http://www.nd.edu/~rwilliam/gologit2